A Novel Loop Filter Design for Phase-Locked Loops

Transcription

1 6 IEEE Coferece o Sytem Ma a Cyberetc October 8-6 ape aa A Novel Loop Flter Deg for Phae-Locke Loop YS Chou WL Mao YC Che a FR Chag Abtract A e loop flter eg metho for phae locke loop (PLL) preete hch employ mult-objectve cotrol techque to eal th the varou eg objectve: mall oe bath goo traet repoe (mall ettlg tme mall overhoot) a large ga a phae marg rae-off amog the coflctg objectve mae va recetly evelope covex optmzato kll cojucto th approprate ajutmet of certa eg parameter Oe alet feature of the propoe metho that t allo oe to pecfy the flter pole avace clug the pecal cae of PI form flter Moreover the propoe metho applcable to PLL of ay orer Numercal mulato o olear PLL moel performe hch emotrate the effectvee of the propoe metho S I INRODUCION INCE t veto the PLL prcple [] [] [3] [4] ha bee ue a e prea of applcato uch a carrer phae trackg [5] tmg recovery [6] a ervo cotrol [7] [8] etc [9] Form the ytem pot of ve PLL eetally a olear ytem he eg of PLL th a uoal phae etector ug Lyapuov reeg techque ca be fou [] [] Hoever there a ffculty applyg the propoe metho to hgh orer loop Recetly a metho bae o lear moel approxmato a propoe hch eg a loop flter that mmze the phae error varace th guaratee ga marg a phae marg the preece of phae etector ga ucertaty a ope loop elay [] Coprme factorzato cotrol theory [3] a quattatve feeback theory (QF) [4] are tegrate to prove a complcate eg proceure I may cae PI form flter hch have all of the pole at the org are favorable for ome avatage [3] they brought : fte hol rage (theoretcally) fte pull- rage (theoretcally) a goo phae trackg capablty Hoever the flter obtae by the approach jut metoe uually have pole ot at the org Havg otce th pot a epeet eg proceure of PI form flter ha alo bee aree [] for eco orer loop Neverthele t ca be checke that t har to exte the metho there to hgher orer loop I [5] the ame problem tue ug aother approach Smlarly the flter obtae are geerally ot of PI YS Chou th the Departmet of Electrcal Egeerg amkag Uverty ape couty aa (phoe: ext 393; fax: ; e-mal: yug@eetkueut) ML Mao th the Departmet of Electrcal Egeerg Mgch Uverty of echology ape couty aa (e-mal: lmao3@ yahoocomt) FR Chag a YC Che are th the Departmet of Electrcal Egeerg Natoal aa Uverty ape aa (e-mal: r9396@tueut) form For global potog ytem (GPS) applcato t ell ko that the moto of the GPS atellte a ell a the GPS recever caue to Doppler effect hch tur reult frequecy hft the carrer a the coe Phae-locke loop are ue to track the gal (carrer a C/A coe) A geerc PLL loop flter eg for GPS recever ca be fou [6 7] here the loop flter obtae are of PI form I th paper bae o the commoly ue voltage-cotrolle ocllator (VCO) moel (a tegrator) e propoe a e loop flter eg metho hch allo oe to pecfy the flter pole avace (clug the pecal cae of PI form flter) Meahle the varou eg objectve: mall oe bath goo traet repoe [8] a large ga a phae marg [8] are all take to coerato rae-off amog them mae va lear matrx equalty (LMI) optmzato [9] [] [] [] cojucto th approprate ajutmet of certa eg parameter he paper orgaze a follo I Secto II the prelmare a problem tatemet are gve I Secto III the e loop flter eg metho preete Secto IV ho the mulato reult Comparo betee the geerc GPS PLL eg a our metho mae Fally Secto V gve the cocluo he etale efto of the H a H orm of table trafer fucto ca be fou [] II PRELIMINARIES AND PROBLEM SAEMEN A Bac moel of PLL u û u Fg PLL chematc moel he PLL moel ue here epcte Fg hch cot of a phae etector lo-pa loop flter F( ) a VCO he put to the phae etector are the to gal: the um of the carrer a bapa oe ( t ) [4] e u( t) = A ( ωt + ( t)) + ( t) a the VCO output uˆ( t) = co( ω ˆ t + ( t)) he phae etector prouce aumg the hgh frequecy term elmate by the lo-pa flter the output gal u ( t) = A [( ˆ ) + ( t)] /6/$ 6 IEEE 9 93

2 here ( t) repreet the et effect caue by the oe ( t ) For mall phae error the PLL ca be further approxmate by the lear moel a epcte Fg W B he goal I ve of Fg the goal of th paper to eg a flter F( ) to acheve the follog objectve: () Cloe-loop tablty () Perfect aymptotcal trackg (e e ( ) = ) ubject to k the etermtc tet gal ( t) = t k = m th = () Goo traet repoe (e mall ettlg tme mall overhoot etc) (v) Noe atteuato (aumg to be hte oe th zero mea) (v) Large tablty marg (e ga marg phae marg) Note that the varace of VCO output phae gve by σ ŵ = ˆ ( jω) ( ) ω ω π Φ here repreet the trafer fucto from ˆ to ŵ a Φ ( ω) the poer pectral ety (PSD) fucto of W the oe If hte Gaua e Φ ( ω) = N the σ ŵ A Ŵ U Fg PLL lear moel approxmato = N ( ) = N B ˆ here B ( Hz ) eote the oe bath of ˆ ( ) It clear that mall oe bath B lea to mall varace of the VCO output phae I term of the loop flter eg th ca be acheve by mmzg the H orm of the cloe-loop trafer fucto ˆ over all the tablzg flter O the other ha the ope-loop repoe houl have eough ga a phae marg orer to guaratee goo relatve tablty a ell-behave cloe-loop repoe It metoe [] [4] that thee marg ca be eterme by the H orm upper bou of certa cloe-loop trafer fucto Specfcally let L( ) eote the ope loop trafer fucto of Fg Aume a cotat γ or alteratvely a cotat δ atfe the follog H orm coto: L( jω) < γ rep < δ () + L( jω) + L( jω) A par of loer bou of the ga a phae marg the eterme by the formula: γ + δ log B; eg rep log B; eg () γ γ δ δ Note that coto () equvalet to a H orm bou cotrat of the cloe-loop trafer fucto hu ŵ mmzg the H orm of the cloe-loop trafer fucto ŵ over all the tablzg flter expecte to creae the ga a phae marg C Geerc GPS PLL eg I the ubecto geerc GPS PLL eg th PI form loop flter trouce [6 7] For eco orer PLL ee Fg 3(a) the loop flter aume to be of the form ω F( ) = aω + Hece the cloe-loop trafer fucto form to ˆ gve by the rato of the Laplace traform of a e W ˆ W W Wˆ aω + ω ( ) = = W + aω + ω he eg parameter a uually choe to be hch yel ampg rato / Furthermore th lea to the follog ueful relatohp: B + a = = 53 ω 4a Smlarly for thr orer loop (th referece to Fg 3(b)) 3 aω ω F( ) = bω Wˆ bω + aω + ω 3 ( ) = = 3 3 W + bω + aω + ω B ab + a b ˆ W 3 ω aω bω ω aω VCO loop flter loop flter = = 7845 f a = b = 4 ω 4( ab ) For fourth orer loop 3 4 bω aω ω F( ) = cω ( a) VCO Fg3 (a) eco orer loop (b) thr orer loop ( b) ˆ 9 933

3 Wˆ cω + bω + aω + ω = = W + cω + bω + aω + ω B c ( ab c) + a( b ac ) bc = = 57 ω 4( abc a c ) f a = 35 b = 65 c = 5 Note that the trafer fucto ŵ of Fg 3 equal to the trafer fucto ŵ or ˆ of Fg he the phae e tector ga A aume to be uty O the other ha mall oe bath B ca be acheve by choog mall value of ω for all the loop Hoever t evet that traet may be acrfce th uch a choce D Ueful Aaly LMI I the follog Lemma [] a Lemma [] preet the H a H orm cotrat of a LI ytem term of LMI repectvely Lemma 3 [] a lemma 4 [] tate that all the egevalue of a quare matrx le a precrbe rego f a oly f certa LMI are feable Lemma []: Gve a LI ytem H( ) = C( I A) B+ D a a potve value ν he follog tatemet are equvalet () D = a the LI ytem H ( ) table a H( ) < ν () D = a there ext matrce P = P a Q = Q uch that the follog LMI hol + AP PA B B < I P ( CP) > CP Q ( ) ν r Q Lemma []: Gve a LI ytem H( ) = C( I A) B + D a a potve valueγ he follog tatemet are equva- let () he LI ytem H ( ) table a H ( ) < γ () here ext a matrx P = P uch that the follog LMI hol AP + PA B ( CP) B γ I D < CP D γ I P > Lemma 3 []: Gve a quare matrx A all the egevalue of A le the coc ector a ho Fg 4(a) f a oly f there ext a matrx X = X uch that the follog LMI are atfe ( )( AX + XA ) co( )( AX XA ) < co( )( XA AX ) ( )( AX + XA ) X > < Lemma 4 []: Gve a quare matrx A all the egevalue of A le the vertcal trp ( h h ) a ho Fg 4(b) f a oly f there ext a matrx X = X uch that the follog LMI are atfe h X AX XA < h X + AX + XA < X > Note that for taar eco-orer ytem a ther approxmato placg the omat pole the coc ector rego (ee Fg 4(a)) th maller reult maller percet overhoot [8] Smlarly placg the omat pole the vertcal trp (ee Fg 4(b)) th larger h reult maller ettlg tme [8] III FIXED-POLE LOOP FILER DESIGN I th ecto a e metho propoe to eg the loop flter for PLL Frtly part A e geeralze the techque [4] to traform the problem of egg a cla of fxe-pole yamc flter to a tatc tate feeback ythe problem Secoly part B mult-objectve tate feeback ythe techque employe to f a loop flter to atfy the eg objectve metoe ecto A Problem reformulato Fg rera a Fg 5 th loop flter F( ) of the form m m+ l f f F( ) = K( ) = + (3) + a + a = = m+ β m here a β are the pole of the flter preeterme Let j ' l = eote the cae that the compoet fucto th pole at β for all are ot coere (3) e m W plae ( a) f F( ) = a m + = If the value of a et to be zero the F( ) of PI form I ve of Fg 5 the gal Y ( ) ca be ecrbe by h h A Y ( ) = W ( ) + W ( ) U ( ) + a plae ( b) Fg4 (a) coc ector (b) vertcal trp ( h h ) W A Y K( ) + a loop flter Û =Ŵ U VCO Fg5 he recotructo moel of the PLL (4) 9 934

4 Defe tate vector ξ ( ) a follo: ξ ( ) = Y ( ) Y( ) Y ( ) Y ( ) Y ( ) m β βl = [ ξ( ) ξ( ) ξ3( ) ξ m + l + ( ) ] a et Z( ) = Uˆ ( ) = U ( ) a Y( ) = ξ ( ) he (4) become ξ ˆ ( ) = aξ ( ) + AW ( ) + AW ( ) AU ( ) Moreover ξ ( ) = ξ ( ) for = 3 m + ξ ( ) = β m ξ ( ) + ξ for = m + m + l + Wth the otato efe above t eay to check that the yamc flter of the precrbe form (3) coverte to a tatc tate feeback the e coorate e Uˆ ( ) = F Y here F [ = f f m + l ] hu the orgal fxe-pole yamc flter eg problem equvaletly traforme to a tatc tate feeback cotrol problem a llutrate Fg 6 th G ecrbe by ξ ( ) = Aξ ( ) + BW ˆ ( ) + BU ( ) ^ GZ( ) = D U ( ) Y ( ) = ξ ( ) here W = W W Z Y a A = β β l m l m l A A ( m+ l+ ) B = [ ] β ( + + ) ( + + ) B = A D = ( m+ l+ ) he reultg cloe-loop ytem from to z a follo ξ ( ) = ( A + BF ) ξ ( ) + BW ( ) z Z( ) = Fξ ( ) For H mmzato from to z (e the oe ba- th mmzato problem) the cloe-loop ytem matrx gve by A + B F B R F G F W ^ U Fg6 he equvalet tate feeback moel here R = [ ] For H mmzato from to z (e the ga/phae marg maxmzato problem) the cloe-loop ytem matrx gve by A + B F B R here R = [ ] F B LMI eg of the loop flter I orer to meet the multple eg objectve metoe ecto the follog mult-objectve cotrol problem coere here α a α are eght for the trae-off of the to eg objectve: oe bath a ga/phae marg Mmze α ν + α γ (5) ( m l ) over N R + + ( m l ) ( m l ) M = M R a Q R atfyg: He( AM + BN ) B R < ( B R ) I (6) M N > N Q race( Q) < ν He( AM + B N) B R N ( B R ) γ I < (7) N γ I M > ( ) He( AM + B N) co( ) SH ( AM + BN ) < (8) { co( ) SH ( AM + BN )} ( ) He( AM + BN ) M > h M He( AM + B N) < (9) h M + He( AM + B N) < M > here the otato He( A) = A + A SH ( A) = A A are ue Deote the optmal oluto by ( N M Q ν γ ) heorem : Gve the oegatve teger l m the value a β β β the eght α l α the potve value a h h th h < h If the optmzato problem (5) olvable there ext a fxe-pole loop flter F ( ) of the form f f F( ) = + + a β th F = [ f fm l ] N M + = uch that () the cloe-loop ytem table m m+ l = = m+ m () ˆ ( ) < ν a ˆ ( ) < γ () the cloe-loop pole le th the terecto of the trp ( h h ) a the coc ector th parameter

5 Proof: he reult follo from applyg taar techque [8] Corollary : Suppoe a = l = a m > here ext a th m -orer loop flter F ( ) of the form m f F( ) = = uch that the reultg cloe-loop ytem poee the properte () () () of heorem hree cae are cue further for corollary Cae : α = = α I th cae oly the factor: oe bath a traet repoe are coere for the loop flter eg h oe va performg the follog H orm mmzato ubject to regoal pole placemet cotrat: m ubject to (6)(8)(9) ν (OP) A algorthm bae o carryg out the optmzato problem (OP) cojucto th approprate ajutmet of the parameter a h preete a follo Algorthm : Gve the ere oe bath B (Hz) a the ere orer m of the loop fler Step Select ampg ratoζ eg ζ 77 e π 4 Step Select h th referece to ABLE Perform (OP) Step 3 If B > B ecreae h ; othere creae h Perform (OP) to get e value of B Cotue th proce utl B = B Step 4 If the traet repoe atfactory the flter eg complete; othere cotue the follog tep: Step 4 If t ot goo eough for overhoot e ecreae a perform (OP) to get B Go to Step 3 Step 4 If t ot goo eough for ettlg tme e creae a perform (OP) to get B Go to Step 3 Note that for practcal tuato the oe bath of the PLL of a GPS recever arou the rage 5~5 Hz [5] hece ABLE ueful electg the parameter h for practcal cae A for the parameter h lttle effect ha bee ABLE SECOND OREDR LOOP ( = π 4 ) h =7 h = h =3 h =6 h =9 B (Hz) HIRD ORDER LOOP ( = π 4 ) h =3 h =5 h =7 h =9 h = B (Hz) FOURH ORDER LOOP ( = π 4 ) h = h =3 h =4 h =5 h =6 B (Hz) fou o the oe bath he t far aay from h Moreover t oberve that t caue to maller oe bath a larger ˆ ( ) he t cloe to h hu the pole cotrat relevat to parameter h omtte cae O the other ha t fou that ecreag mprove overhoot; hoever the oe bath creae I orer to mata the ame oe bath the parameter h ecreae But th lea to a larger ettlg tme hu there a trae-off betee oe bath a ettlg tme Smlarly there a trae-off betee oe bath a overhoot Cae : α = α = I th cae the follog optmzato problem performe for the loop flter eg to maxmze the ga/phae marg a eure goo traet repoe m γ (OP) ubject to (7)(8)(9) he follog relatohp betee the H orm a the parameter h h a are explore: the H orm rectly proportoal to h a verely proportoal to h But the oe bath rectly proportoal to h th cae Moreover t a oberve that performg (OP) thout the LMI cotrat cocerg h uually caue ome cloe-loop pole to be far aay from the magary ax h tur lea to large oe bath hu the pole cotrat relevat to parameter h clue th cae for the ajutmet of the oe bath Cae 3: α aα are potve umber h cae a mxture of the former to cae Noe bath ga/phae marg a traet repoe are all take to coerato Wthout lo of geeralty α may be aume to be uty he loop flter eg oe va performg the follog optmzato problem cojucto th approprate ajutmet of the parameter a h h : m ν + α γ ubject to (6)(7)(8)(9) (OP3) Large value for eght α (e α > ) choe f the ga /phae marg are emphaze O the cotrary oe atteuato emphaze by electg a mall eght forα (e α < ) he parameter h h a are tue th referece to Cae a Cae Remark I the geerc GPS PLL eg there a cloe form formula for the rato B ω term of the eg para- meter for example the parameter a b a c for the fourth orer loop (ee Secto C) After etermg the parameter through mmzg the rato B ω the rato keep cotat A a reult the oe bath proportoal to the value of ω Smaller oe bath requremet lea to maller value of ω hch caue ome of the cloe-loop

6 pole to move toar the magary ax ce the charactertc polyomal for the fourth orer loop gve by cω + bω + aω + ω hece lo o the ytem repoe Clearly t har to meet the multple objectve a metoe Secto by the gle parameter ω I comparo our metho prove a flexble a geeral ay to yel a flter va trag off the objectve through optmzato over all the eg parameter f IV SIMULAION RESULS I the practcal tuato the ere oe bath of PLL of GPS recever electe to be arou 5 Hz e B 5Hz hu th eg objectve eforce for all the cae cue here By the geerc GPS PLL eg ecrbe Secto C PI form flter of fferet orer (e m = 3 ) are obtae oce the target oe bath B gve o apply our metho (corollary ) thout lo of geeralty the phae etector ga A aume to be uty ABLE ho the trafer fucto of the loop flter obtae by the geerc GPS PLL eg a our LMI approach repectvely ABLE (A) GENERIC GPS PLL DESIGN Loop orer Deg parameter Loop flter F( ) a = Several performace ce are evaluate for the reultg PLL a ho ABLE 3 For eco orer loop the geerc eg a ear optmal eg [3] It oberve that our approach obta a flter hch reult mlar performace Moreover for hgher orer loop by our approach there are gfcat mprovemet o the traet performace a ell a the ga/phae marg over the geerc eg Note that the reultg PLL are cotoally table ytem a efe [4] hch have egatve ga marg he ga a phae marg o the lt are the guaratee value compute by the formula () hch mply that each of the cloe loop ytem rema table f the ope loop ga ecreae le tha the guaratee value Next e ue the popular oftare SytemVe Vero 5 to mulate the reultg PLL A olear ytem moel a ho Fg 7 ue here the fucto of the phae ' 4 + ω = 83 a = b = ω = 9 a = 35 b = 65 c = ω = 49 ABLE (B) LMI APPROACH Loop orer Deg parameter Loop flter F( ) α = α = = π 4 h = 97 3 α 4 α = α = 35 = π 6 h = 46 h = 64 = α = 3 = π 6 h = 4 h = ABLE 3 he performace ce for oe bath 5 Hz Deg Loop Re tme Overhoot Settlg H metho orer (ec) (%) tme orm (ec) LMI approach Geerc GPS etector moele a A ( ) tea of a cotat ga A Deg metho Loop orer Ga marg Phae marg 57 B 4676 eg 3 58 B 494 eg 4 56 B 495 eg 54 B 4637 eg Geerc GPS 3 4 B 34 eg 4 6 B 79 eg ŵ A ( ) hr orer a fourth orer PLL loop are mulate for tetg the lock- performace heoretcally the lock rage ω L efe a the frequecy rage th hch the PLL lock th oe gle beat ote betee the referece frequecy a VCO output frequecy [3] It ca be etmate by olvg the follog equato for [3] ω = A F( j ω ) L L For thr orer loop the etmate lock rage ω are L 488 (ra/) a 436 (ra/) for a geerc eg repectvely Frequecy chage th a over both of the lock rage equvaletly a phae ramp put are apple to the loop he repoe are ho Fg8 (a) a (b) It oberve that the PLL ege by both approache get locke But the traet repoe by LMI eg look better tha that by the geerc eg Fally ame tet apple to the fourth orer loop he reult ho Fg 9 (a) a (b) Aga the PLL eg by behave better tha that by geerc eg Partcularly for hgher frequecy chage hle the PLL by geerc eg loe locke that by tll ork very ell a ca be ee Fg 9(b) V CONCLUSION loop flter VCO Fg 7 he mulato moel ug SytemVe We have preete a e flter eg metho for PLL takg to coerato the varou eg objectve uch a mall oe bath goo traet repoe (mall ettlg tme mall overhoot) a large ga a phae marg I comparo th everal extg metho the propoe ωl

7 metho mple a applcable to PLL of ay orer Partcularly t allo oe to pecfy the flter pole to ere locato avace (clug the pecal cae of PI form flter hch have all the pole at the org) Numercal mulato of a GPS applcato a performe ug olear PLL moel It a oberve that our metho yel much better performace he compare th the tratoal GPS PLL eg frequecy (ra/ec) frequecy (ra/ec) tme (ec) (a) ACKNOWLEDGMEN geerc eg geerc eg tme (ec) (b) Fg 8 he frequecy of VCO output of thr orer loop for phae ramp put (a) 4 a (b) 85 ( ra ) frequecy (ra/ec) frequecy (ra/ec) geerc eg tme (ec) (a) geerc eg tme (ec) (b) Fg9 he frequecy of VCO output of fourth orer loop for phae ramp put (a) 35 a (b) 85 ( ra ) h ork a upporte part by Natoal Scece Coucl uer Grat NSC 94-3-E--79 Grat NSC 94-3-E-3-9 a Grat NSC 9-3-E-3-8 REFERENCES [] A Blachar Phae-Locke Loop: Applcato to Coheret Recever Deg Wley: Ne York 978 [] A J Vterb Prcple of Coheret Commucato McGra-Hll: Ne York 966 [3] R E Bet Phae-locke loop: Deg Smulato a Applcato McGra-Hll Iteratoal997 [4] F M Garer Phaelock echque Joh Wley Ne York NY thr e 5 [5] A Abu-Rgheff MN Sumartaa IGG Carrer phae trackg gtal rao commucato Electroc Letter vol 34 pp [6] A De Glora D Groo M Olver a G Reta A ovel tablty aaly of a PLL for tmg recovery har k rve IEEE ra Crcut a Sytem-I: Fuametal heory a Applcato vol 46 o 8 pp [7] M F La M Nakao a GC Heh Applcato of Fuzzy logc the phae-locke loop pee cotrol of ucto motor rve IEEE ra o Iutral Electroc vol 43 o6 pp [8] J W Ah S G Oh S Y Pyo CU Km a Y M Hag Dgtal PLL techque for prece pee cotrol of SR Drve Poer Electroc Specalt Coferece ( PESC99) 3th Aual IEEE vol 999 pp [9] G C Heh a J C Huag Phae-Locke Loop echque- A Survey IEEE ra o Iutral Electroc vol 43 o6 pp [] D Y Abramovtch Aaly a eg of a thr orer phae-lock loop Proceeg of the IEEE Mltary Commucato Coferece vol October 988 pp [] D Y Abramovtch Lyapuov Reeg of aalog phae-lock loop IEEE ra o Commucato vol 38 pp 97- December 99 [] O Yav D Raphael Near-optmal PLL eg for eco feeback carrer a tmg recovery IEEE ra o Commucato Vol 49 pp Sept [3] M Vyaagar Cotrol Sytem Sythe-A Factorzato Approach MI pre 985 [4] O Yav Quattatve Feeback Deg of Lear a Nolear Cotrol Sytem Norell MA: Kluer 999 [5] V Supl a U Shake Robut H-fty cotrol of phae-locke loop th polytopc type ucertate It J Robut a Nolear Cotrol vol pp [6] ED Kapla Uertag GPS: prcple a applcato Artech Houe Loo 996 [7] W L Mao H W ao a F R Chag Itellget GPS recever for robut carrer phae trackg kematc evromet IEE Proceeg Raar Soar a Navgato vol 5 No 3 pp7-8 Ju 4 [8] N S Ne Cotrol Sytem Egeerg Joh Wley & So Ic 4 th eto 4 [9] S Boy L EL Ghaou E Fero a V Balakrha Lear Matrx Iequalte Sytem a Cotrol heory SIAM Phlaelpha 994 [] C Scherer P Gahet a M Chlal Multobjectve output feeback cotrol va LMI optmzato IEEE ra Autom Cotrol vol 4 pp [] M Chlal a P Gahet H eg th pole placemet cotrat: a IEEE ra Autom Cotrol vol 4 pp [] P Gahet A Nemrovk A Laub a M Chlal LMI Cotrol oolbox he MathWork Ic Natck MA 995 [3] K Zhou a J C Doyle Eetal of Robut Cotrol Pretce-Hall Ic Ne Jerey 998 [4] C E e Souza a U Shake A LMI metho for output-feeback H cotrol eg for ytem th parameter ucertaty Proc IEEE Cof Deco a Cotrol ampa Flora USA pp [5] M S Braach a A J Va Dereock GPS recever archtecture a meauremet Proceeg of the IEEE vol 87 pp